IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 10, OCTOBER

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1 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 10, OCTOBER High-Frequency Modeling of the Long-Cable-Fed Induction Motor Drive System Using TLM Approach for Predicting Overvoltage Transients Liwei Wang, Student Member, IEEE, Carl Ngai-Man Ho, Member, IEEE, Francisco Canales, Member, IEEE, and Juri Jatskevich, Senior Member, IEEE Abstract Induction motor drive systems fed with cables are widely used in many industrial applications. Accurate prediction of motor terminal overvoltage, caused by impedance mismatch between the long cable and the motor, plays an important role for motor dielectric insulation and optimal design of dv/dt filters. In this paper, a novel modeling methodology for the investigation of long-cable-fed induction motor drive overvoltage is proposed. An improved high-frequency motor equivalent circuit model is developed to represent the motor high-frequency behavior for the timeand frequency-domain analyses. The motor equivalent circuit parameters for the differential mode (DM) and common mode (CM) are extracted based on the measurements. A high-frequency cable model based on improved high-order multiple-π sections is proposed. The cable model parameters are identified from the DM impedances in open circuit (OC) and short circuit (SC). To obtain a computationally efficient solution that could potentially be integrated with the motor drive controller, the system equations are discretized and solved using transmission-line modeling (TLM) approach. The proposed methodology is verified on an experimental 2.2-kW ABB motor drive benchmark system. The motor overvoltage transients predicted by the proposed model is in excellent agreement with the experimental results and represents a significant improvement compared with the conventional models. Index Terms High-frequency modeling, impedance measurement, induction motor drives, long cable, overvoltage transients, transmission-line modeling (TLM). I. INTRODUCTION THE ADJUSTABLE speed drive systems are undergoing significant changes with the successful application of novel power semiconductor devices. The rise and fall switching times of newer devices have been decreased in order to reduce switching losses and increase the overall efficiency. However, very fast switching times coupled with the long cables can cause voltage reflection at the motor terminals due to cable motor surge impedance mismatch [1], [2]. In some cases, the motor terminal overvoltage can reach two times (or even higher) of the Manuscript received October 6, 2009; revised February 12, 2010; accepted March 12, Date of current version September 17, Recommended for publication by Associate Editor B. Ferreira. L. Wang and J. Jatskevich are with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T1Z4, Canada. C. N.-M. Ho and F. Canales are with the ABB Corporate Research Center, Baden-Daettwil 5405, Switzerland ( carl.ho@ch.abb.com). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TPEL Fig. 1. Typical motor drive system with connecting cable. dc-bus voltage [3], [4], which may seriously harm the motors dielectric insulation leading to subsequent failures [5], [6]. Therefore, accurate modeling of the cable-fed motor drive systems in the high-frequency range is crucial for predicting the motor terminal overvoltage and designing mitigation filters [7] [10]. A typical system composed of a drive feeding an induction machine through a cable is shown in Fig. 1. In order to facilitate the high-frequency cable and motor modeling and characterization, two modes of operations, common mode (CM) and differential mode (DM) [11], are usually considered. The CM is formed as a two-port network between the parallel-connected three phases and ground wire. The DM is regarded as a two-port network between two parallel-connected phases and the third phase of the cable motor system. Numerous high-frequency motor models have been proposed in the literature for overvoltage and electromagnetic interference (EMI) analysis [12] [18]. A pioneering basic phase-belt winding model was proposed in [12] to represent motor singlephase coils. Therein, a high-frequency model was established based on series connections of four basic phase-belt windings. Subsequent research [13] [18] on high-frequency motor models has focused on constructing more easy-to-use motor equivalent circuits and straightforward parameter identification procedures. One [13], [15], [17] or two [14], [16] phase-belt winding motor models are proposed, which greatly simplifies the highfrequency motor model. In [17], an efficient high-frequency motor model is proposed for the studies of both DM and CM behaviors in time- and frequency-domain. Therein, a simple and easy-to-follow model parameterization procedure is included based on DM and CM impedance measurements. In [18], a universal induction machine model with low-to-high frequencyresponse characteristics is proposed. This model is quite sophisticated as the DM, CM, and bearing models are all integrated into one three-phase motor equivalent circuit /$ IEEE

2 2654 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 10, OCTOBER 2010 Modeling of long cables for analysis of motor voltage reflection phenomena has been studied as well [7] [9], [19] [23]. It has been recognized in [15] and [20] [23] that a realistic and accurate cable model is of great importance for the accurate prediction of motor terminal overvoltage. Various cable models were proposed including conventional multiple π-section model with second-order per section [7], [8], lossless [9], [19], lossy [20] [22], and frequency-dependent [15], [23] transmission-line models (TLMs). In [21], a lossy but distortionless cable model is used, which was shown to improve the accuracy of the predicted overvoltage compared with the lossless cable model. In [15], a multiple-π cable model was proposed, where the cable parallel branch is represented as a high-order capacitive circuit to include the frequency-dependent dielectric losses. Oriti and Julian [23] represented the entire cable motor system in frequency domain to model the frequency-dependent effects of the cable and motor. However, to the best of our knowledge, a time-domain frequency-dependent cable model, which includes both the skin and proximity effects and the dielectric losses for the high-frequency range, have not been developed for the motor overvoltage studies. This paper proposes such a model and makes the overall contributions that can be summarized in the following. 1) An improved motor model, based on [17], is proposed that accurately captures the high-frequency DM and CM impedance characteristics from hundreds of hertz to tens of megahertz. We also provide detailed and easy-toreproduce parameter identification and characterization procedure for the improved equivalent circuit model. 2) A more accurate frequency-dependent cable model is developed using a higher-order multiple π-sections, which includes the skin and proximity effects as well as dielectric losses. The model parameters are identified based on the measured DM impedance characteristics in a wide frequency range from 100 Hz to 10 MHz. 3) For efficient calculation of the time-domain transient solution, the continuous-time cable motor system is discretized using the TLM approach [24] [27]. Therefore, the formulation and integration of differential equations is avoided, and the entire cable motor system (with arbitrary number of cable sections) could be implemented in a stand-alone program and hardware for control algorithms and/or firmware applications. 4) The proposed methodology is verified experimentally as well as compared against two conventional models [7], [8]. It is shown that the proposed model represents an appreciable improvement in predicting the motor overvoltage transients. II. HIGH-FREQUENCY INDUCTION MOTOR MODELING Accurate modeling of induction motors in high-frequency range plays an important role in investigating the motor drive overvoltage and EMI problems [12] [18]. In many cases, motor equivalent circuit and the associated parameters are established from the motor impedance characteristics and/or geometry. The high-frequency motor model used in this paper is based on per- Fig. 2. High frequency per-phase motor equivalent circuit (a) model defined in [17] and (b) proposed model. phase equivalent circuit proposed in [17]. The circuit diagram of the motor per-phase high-frequency model is illustrated in Fig. 2(a). This circuit provides relatively simple model structure. However, the parameters R ad and C ad in Fig. 2(a) are very difficult to express using analytical formulas [17]. Instead, trials and adjustments of R ad and C ad are usually required with the help of frequency-domain simulations to achieve satisfactory results [17]. In this paper, an improved high-frequency model is proposed as shown in Fig. 2(b). The proposed circuit uses similar model structure and parameterization procedure as in [17]. However, a series R t, L t, and C t branch is introduced [see Fig. 2(b)] to replace the R ad and C ad branch [see Fig. 2(a)] without sacrificing model accuracy. Since this new series branch has its own resonance, analytical calculation of parameters R t, L t, and C t can be carried out in a more straightforward way from the measured impedances, which represents an advantage over identification of the R ad and C ad branch in Fig. 2(a). The detailed derivations for the branch parameters are not included here due to limited space, whereas interested reader can find more background information in [13] and [15]. The equivalent circuit depicted in Fig. 2(b) and its parameters also have physical meaning and significance. In particular, R g1 and C g1 represent the parasitic resistance and capacitance between the stator winding and the motor frame; R g2 and C g2 represent the parasitic resistance and capacitance between the stator neutral and motor frame; L d represents the stator winding leakage inductance; R e represents the high-frequency iron loss of the stator winding; L t and C t are introduced to capture the second resonance in the motor impedance characteristic, which may be caused by the skin effect and interturn capacitance of the stator windings.

3 WANG et al.: HIGH-FREQUENCY MODELING OF THE LONG-CABLE-FED INDUCTION MOTOR DRIVE SYSTEM 2655 TABLE I HIGH-FREQUENCYMOTOR PARAMETERS Fig. 3. Measured DM and CM impedances of the induction motor under study. The overall motor high-frequency three-phase equivalent circuit model can be obtained assuming a wye connection of three single-phase circuits of Fig. 2(b). Such assumption is widely used in high-frequency machine modeling [13] [18], and it has an advantage of simpler model structure and easier identification of the associated parameters. In particular, the parameters of the per-phase equivalent circuit in Fig. 2(b) are identified through the DM and CM impedance characteristics measured in frequency domain. However, it is also noted that the proposed three-phase high-frequency motor model is valid independent of the actual physical connection of the stator windings (i.e., deltaor wye connection). In other words, whether the actual stator winding is delta- or wye-connected, the procedure presented in this paper will give an equivalent wye-connected circuitmodel that will adequately represent the considered overvoltage phenomena. The motor under investigation is the four-pole, deltaconnected, 2.2-kW, ABB M2AA100LA. The detailed DM and CM measurements setup and test procedures have been well documented in the literature, e.g., [17]. Similar measurement procedures have been used here. In particular, the motor impedances have been measured when the motor is stationary and disconnected from the ac source (drive) [13] [18]. As it has been shown in [18] and [28], the rotor speed or the stator current has no significant influence on the motor impedance characteristics in mid-to-high frequency range. The underlying reason is that the flux penetrating into the rotor magnetic circuit at high frequency is very small [28], [29]. For the measurements presented here, an HP/Agilent 4294 A impedance analyzer [30] is used to evaluate the impedances in the range from 100 Hz to 10 MHz. The measured DM and CM impedances Z DM and Z CM are shown in Fig. 3, respectively. To calculate the parameters of the high-frequency motor model, several characteristic points are chosen from the measured DM and CM impedances of Fig. 3. First, the low- and high-frequency CM capacitances (slopes) C total and C HF are extracted from the measured CM impedance Z CM, respectively. Second, the resonance frequency and impedance magnitude of the first zero point (Z 1 ) from Z CM are identified. Third, the lowfrequency DM inductance L DM is evaluated from the measured DM impedance Z DM. Finally, the resonance frequencies and impedance magnitudes of the pole (P ), the second (Z 2 ), and the third (Z 3 ) zero points are measured from Z DM. Given these characteristic measurements, the parameters of the high-frequency motor model in Fig. 2(b) are readily calculated according to the equations summarized in Appendix A. For example, the parameters R t, L t, and C t are derived from the second resonant point (Z 2 ) in the DM impedance Z DM characteristic shown in Fig. 3. The corresponding formulas are given in Appendix A (A8) (A10) for consistency. The resulting high-frequency motor parameters for the considered induction motor are summarized in Table I. To verify the effectiveness of the proposed equivalent circuits and the procedure for calculating parameters, the measured DM and CM impedances are compared to the corresponding impedances calculated from the high-frequency motor equivalent circuits. The high-frequency DM motor equivalent circuit is constructed here by connecting two of the phases in parallel and the remaining third phase. Based on the per-phase equivalent circuit in Fig. 2(b), the combined DM motor equivalent circuit is shown in Fig. 4(a). The high-frequency CM motor equivalent circuit is built by connecting all the three phases in parallel. Based on per-phase equivalent circuits of Fig. 2(b), the resulting CM motor equivalent circuit from phase to ground is shown in Fig. 4(b). The measured and fitted results using the constructed equivalent circuits are superimposed in Fig. 5 for better clarity. As can be observed in Fig. 5, the fitted DM and CM frequency characteristics have good agreement with the measured responses. These results verify the effectiveness of the proposed three-phase wye-connected equivalent circuit model for representing the high-frequency characteristics of the motor windings (although the stator winding is delta-connected). It is noted that the motor line-to-line overvoltage ringing transient is mostly associated with DM, since this mode defines the physical path along which the insulated gate bipolar transistor (IGBT)-generated voltage pulse travels through the cable and the stator winding of the motor. Since the line-to-line overvoltage predominantly impacts the insulation of windings and cables, this type of overvoltage is primarily important for the purpose of this paper. Therefore, the DM equivalent circuit in Fig. 4(a) is used here for motor drive overvoltage studies. III. HIGH-FREQUENCY CABLE MODELING Accurate modeling of power cable is also essential for predicting motor terminal overvoltage. Since the overvoltage transient

4 2656 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 10, OCTOBER 2010 Fig. 4. High-frequency equivalent circuit motor model: (a) for DM assuming phases B and C are connected in parallel and (b) for CM between the three connected phases and ground. at the motor end may contain oscillations in wide range of frequencies from several hundreds of hertz to several megahertz depending on the cable characteristic and length, the cable model should also include the frequency-dependent phenomena such as skin and proximity effects, dielectric losses. The available methods for modeling cables at this level and extracting parameters include finite-element analysis [22] or direct impedance measurement [15], [31] in frequency domain using precision impedance analyzers. In this paper, the same impedance analyzer was also used to characterize the cable. The cable used here is an unshielded, PVC-insulated, fourcore cable with the conductor area of 1.5 mm 2. The short circuit (SC) and open circuit (OC) tests in DM [22], [31] have been performed upon which the cable parameters were calculated. The measured shunt capacitance and conductance (OC test), and the series resistance and inductance (SC test) for 1-m sample cable section are also shown in Fig. 6. As can be observed in Fig. 6, the series resistance and inductance, as well as the parallel conductance and capacitance, all appear strongly frequency-dependent, especially in the range of high frequencies. However, this phenomenon is often ignored in most of the published literature on the motor drive overvoltage studies. When such assumption is made, the resulting per-section equivalent circuit is just a conventional second-order per-section cable model shown in Fig. 7(a) [7], [8]. The advantage of equivalent circuit of Fig. 7(a) is of course its simplicity. The parameters for this circuit are typically identified at the resonant frequency of motor terminal overvoltage. Fig. 5. Measured and fitted motor impedances for (a) DM and (b) CM. However, the resonant frequency itself will also depend on the cable length and may be difficult to predict with good accuracy. An improved cable model was proposed in [15], which can represent the dielectric losses using a higher order parallel branch as shown in Fig. 7(b). Further improvements are proposed in this paper in order to include the skin and proximity effects. Such effects are not taken into account with the use of first-order series branch. Therefore, a high-order model is proposed in Fig. 7(c). This section model combines the parallel branches as proposed in [15] and utilizes higher order series branches to represent skin and proximity effects. Overall, one additional inductor and

5 WANG et al.: HIGH-FREQUENCY MODELING OF THE LONG-CABLE-FED INDUCTION MOTOR DRIVE SYSTEM 2657 Fig. 6. Frequency-dependent parameters of the cable as measured in DM. Similarly, based on the results of Fig. 9 in OC Z OC, the corresponding capacitances can be determined from the 20-dB/decade slope in the low- and high-frequency regions as C p1 + C p2 = C OC-LF (3) C p1 = C OC-HF (4) where C OC-LF and C OC-HF are the low- and high-frequency open-circuit cable capacitance shown in Fig. 9, respectively. The aforementioned procedure readily establishes parameters of the equivalent circuit section of Fig. 7(c). More detailed formulas for cable parameter calculation are given in Appendix B and the resulting parameters are summarized in Table II. To show the effectiveness of this procedure, the measured and the fitted SC and OC impedances are superimposed in Fig. 10. As can be observed in Fig. 10, the results produced by the proposed equivalent circuit of Fig. 7(c) are in good agreement with the measured characteristics for the actual cable segment. It is observed in Fig. 10(b) that the absolute fitting error of the phase angle θ OC is within ±5 starting from 400 Hz to MHz range (which is within the targeted frequency range of the equivalent circuit for motor overvoltage studies). capacitor are utilized in the per-section model compared with the classic second-order constant parameter cable model. To identify the parameters of the circuit of Fig. 7(c), the circuit response is considered in the low- and high-frequency ranges, respectively. In particular, in the low-frequency range, the SC and OC circuits would appear as depicted in Fig. 8(a) and (c), respectively. Similarly, for high-frequency range, the SC and OC circuits will appear as in Fig. 8(b) and (d), respectively. The separation into low- and high-frequency ranges facilitates the parameter identification procedure for the proposed section circuit of Fig. 7(c) from the measured impedances. To illustrate this point, the measured SC and OC impedances Z SC and Z OC of the 1-m cable section in DM are depicted in Fig. 9. As shown in Fig. 9, the measured SC impedance Z SC can be approximated over low- and high-frequency ranges by the first-order inductance transfer functions with the asymptotes of 20-dB/decade slope. The low- and high-frequency SC inductances L SC-LF and L SC-HF are directly extracted from the magnitude of Z SC.As it can be observed in Fig. 9, the inductance value of L SC-HF is smaller than L SC-LF. This difference is attributed due to the skin and proximity effects, which again verifies the frequencydependent nature of the measured cable impedance, which is represented by the series branch in the equivalent circuit. A similar conclusion applies to the measured OC cable impedance Z OC as shown in Fig. 9. Based on SC results Z SC, the corresponding inductances can be determined from the 20-dB/decade slope in the low- and highfrequency regions, L SC-LF and L SC-HF, using the following relationships: L s1 + L s2 = L SC-LF (1) L s1 = L SC-HF. (2) IV. TLM FORMULATION OF THE LONG-CABLE-FED MOTOR DRIVE SYSTEM In this paper, the TLM approach is used to formulate the difference equations of the overall long-cable-fed motor drive system for time-domain simulation as a stand-alone program. This approach also provides a convenient means to consider arbitrary number of cable sections for accurate representation of cables of different lengths. Depending on component characteristics, stub lines with different termination conditions are utilized to represent various circuit components. For linear resistors, the algebraic equation given by Ohm s law is used in TLM representations. The inductor TLM model is usually represented as a Thevenin equivalent circuit as shown in Fig. 11(a). For an inductor L, a short-circuited stub line is used, given the surge impedance of Z L =2L/ t, where t is the simulation time step and also round-trip traveling time along the stub line. The inductor branch voltage equation is then formulated as [24] [27] v L (t) =Z L i L (t)+2v i L(t) (5) where vl i (t) is the incident voltage pulse at the present time step t. For the calculation purpose, vl i (t) is required to be represented by the branch voltage and incident voltage pulse at the previous time step, e.g., v L (t t) and vl i (t t). Forthe short-circuited stub, the reflected voltage pulse at the previous time step vl r (t t) is inverted when it travels back from the line end after the round-trip traveling time t. Therefore, the incident voltage pulse at the present time step vl i (t) is represented as v i L(t) = v r L(t t) =v i L(t t) v L (t t). (6) Similarly, a capacitive branch is modeled as an open-circuited stub line depicted in Fig. 11(b) with surge impedance of Z C = t/2c. The capacitor is also represented as a Thevenin

6 2658 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 10, OCTOBER 2010 Fig. 7. Per-section DM cable models. (a) Conventional. (b) improved model [15]. (c) Proposed model. Fig. 8. SC and OC cable model representations as viewed in low- and high frequencies. Fig. 9. Measured SC and OC impedances of the 1-meter cable section in DM. Fig. 10. Measured and fitted cable impedances. (a) SC. (b) OC. TABLE II PROPOSED CABLE PER-METER PARAMETERS equivalent circuit shown in Fig. 11(b). The branch voltage equations is given as v C (t) =Z C i C (t)+2vc i (t). (7) As the capacitive stub is open circuited, the reflected voltage pulse at the previous time step vc r (t t) travels back without inversion and becomes the incident voltage pulse at the present

7 WANG et al.: HIGH-FREQUENCY MODELING OF THE LONG-CABLE-FED INDUCTION MOTOR DRIVE SYSTEM 2659 Fig. 13. TLM representation of the per-section DM cable model. (a) Original circuit. (b) Thevenin equivalent. Fig. 11. TLM models of (a) inductance and (b) capacitance. Fig. 14. Final formulation of DM long-cable motor drive models in discrete time domain. R s + R p R p R p R s +2R p R p Z =. R.. p Rp R p R s +2R p R p R p R p + R eq Fig. 12. TLM representation of the DM motor model. (a) Original circuit. (b) Reduced Thevenin equivalent. time step after the round-trip traveling time t as v i C (t) =v r C (t t) =v C (t t) v i C (t t). (8) In order to represent the DM motor and per-section cable models using TLM technique, the inductances and capacitances in Figs. 4(a) and 7(c) are replaced by their corresponding impedances and incident voltage sources. The resultant TLM representations of the DM motor and per-section cable model are formulated in Figs. 12(a) and 13(a), respectively. For the purpose of convenience, the Thevenin equivalent circuits of Figs. 12(a) and 13(a) are calculated and are given as Figs. 12(b) and 13(b), respectively. Therein, e eq (t t), e s (t t), and e p (t t) are dependent on the previous time step as the incident voltages in Figs. 12(a) and 13(a) are calculated by (6) and (8), respectively. Due to space limitation, the detailed expressions for the Thevenin impedances and voltage sources in Figs. 12(b) and 13(b) are not included (these are easily calculated using network reduction techniques). Assuming an arbitrary number of cable sections, the entire DM long-cable motor drive system in discrete time domain is represented in Fig. 14. At each time step, the loop current equations are then used to solve the system of Fig. 14 as where ZI = V (9) (10) V =[v inv (t) e s1 (t t) e p1 (t t) e p1 (t t) e s2 (t t) e p2 (t t) e p,n 2 (t t) e s,n 1 (t t) e p,n 1 (t t) e p,n 1 (t t) e eq (t t)] T (11) I =[i 1 (t) i 2 (t) i n 1 (t) i n (t)] T. (12) Here, Z is constant system resistance matrix; V is the loop voltage vector; I is the unknown loop current vector. It is noted that Vis known for the present time step as the Thevenin equivalent voltage sources in (11) are known from the previous time step and the inverter voltage v inv (t) is given. Since the matrix Z is tridiagonal, sparse matrix technique may be applied to solve the system equations very efficiently. At each time step, the loop currents are solved first from (9). The branch currents and voltages are then calculated according to Figs. 12(a) and 13(a). The incident voltages are finally updated using (6) or (8), depending on the inductive or capacitive branches, for the next time step. V. EXPERIMENTAL VERIFICATION To validate the methodology described in this paper, the experimental tests with long-cable-fed motor drive system have been carried out and are presented in this section. The drive system is ABB 2.2-kW ACS50-type with switching frequency of 16 khz. The rise and fall times of IGBT switches are approximately 190 µs. The power cable is an unshielded, PVC-insulated, four-core cable with the conductor area of

8 2660 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 10, OCTOBER 2010 TABLE III CONVENTIONAL CABLE PER-METER PARAMETERS (a) AT 1kHz.(b)AT 0.4 MHz Fig. 15. Measured voltage waveforms at the inverter and motor terminals for 100-m cable. 1.5 mm 2. The same four-pole, 2.2-kW, ABB (model number: M2AA100LA) induction machine was used in all tests. The measured impedances of the cable and the motor have been illustrated in Sections II and III. In order to demonstrate the importance of including frequency-dependent effects in cable modeling, two conventional models and the proposed model have been considered. All considered models use multi-π sections equivalent circuits to approximate the distributed parameter effects, wherein the length of each cable section is assumed to be 0.1 m. The conventional models use second-order per-section equivalent circuit shown in Fig. 7(a) with parameters R s, L s, R p, and C p measured at low- and high-frequency (1 khz, Conventional model 1) and high-frequency (0.4 MHz, Conventional model 2), respectively. For consistency, the corresponding parameters are summarized in Table III. The high-frequency measurement at 0.4 MHz is chosen because the overvoltage oscillations for the considered cable were in the same frequency range. The proposed fourthorder per-section cable model has also been used. The model parameters were identified according to the experimental procedure described in Section III and results are summarized in Table II. For investigation of motor overvoltage transients, the cable lengths of 20, 25, 50, 60, 70, 100, and 120 m have been studied. To demonstrate the overvoltage phenomena on the considered motor drive system with a 100-m cable, the measured line-toline voltages at the inverter side and the motor terminal are shown in Fig. 15. As can be seen in this figure, the rectangular pulses generated by the inverter switching produce oscillatory transient and significant overvoltage at the motor terminal. A fragment of Fig. 15 has been magnified and superimposed with the simulated responses produced by the proposed and conventional models as depicted in Fig. 16(a) and (b), respectively. As can be seen in Fig. 16(a), the proposed model predicts more accurately both the waveform and the damping than either of the conventional models in Fig. 16(b). The models are further compared to the experimental results for the 70- and 120-m cables, as shown in Figs. 17 and 18, respectively. It is observed in Figs. 16(b), 17(b), and 18(b) that the conventional models 1 and 2 either over- or under estimate the overvoltage peaks. Also, since the attenuation and distortion effect due to frequency de- Fig. 16. Measured voltage waveforms at the motor terminals for 100-m cable. (a) Proposed model. (b) Conventional models 1 and 2. pendence are not represented, the shape of the overvoltages predicted by models 1 and 2 are also more square-like. The overall damping time of the overvoltage transient predicted by the model 1 is also much longer compared with model 2. This also clearly shows that using the second-order section model with either low- or high-frequency parameters (see Table III) is not adequate to cover the spectrum of the overvoltage phenomena. At the same time, the proposed model demonstrates good results in predicting the overvoltage peaks, shape, as well as damping as can be clearly seen in Figs. 16(a) 18(a). To further evaluate the considered models in predicting the overvoltage transient phenomena, the cable delay time t p,oscillation frequency f osc, and peak overvoltage have been compared with the experimental results for different cable lengths. The results of these comparisons are summarized in Figs The delay time t p is defined with respect to the initial pulse produced by the inverter as shown in Fig. 16(a). The oscillation frequency f osc is defined for the voltage on the motor terminal also shown

9 WANG et al.: HIGH-FREQUENCY MODELING OF THE LONG-CABLE-FED INDUCTION MOTOR DRIVE SYSTEM 2661 Fig. 19. Measured and predicted delay times for different cable lengths. Fig. 17. Measured voltage waveforms at the motor terminals for 70-m cable. (a) Proposed model. (b) Conventional models 1 and 2. Fig. 20. lengths. Measured and predicted oscillation frequencies for different cable Fig. 18. Measured voltage waveforms at the motor terminals for 120-m cable. (a) Proposed model (b) Conventional models 1 and 2. in Fig. 16(a). Both t p and f osc can also be expressed as [20] t p = l L s C p (13) 1 f osc = 4l (14) L s C p where l is cable length, L s and C p are equivalent per-meter inductance and capacitance for a specific traveling wave frequency. As seen in Fig. 19, the delay times are almost linear, which is consistent with (13). The conventional model 1 has a higher slope than the measured delay times because it relies on pa- rameters extracted at low frequency. The conventional model 2 gives more accurate delay times, since its parameters were extracted at high frequency that is closer to the ringing frequency. However, the proposed model predicts delay times with an even greater accuracy. Figs. 20 and 21 show the oscillation frequencies and the peak of the line-to-line overvoltage as measured from the drive-cable-motor system and predicted by various models. A similar consistent conclusion applies here. The lower order conventional models 1 and 2 are less accurate and the proposed model predicts the overvoltage phenomena in terms of both the oscillation frequencies and the peak of the line-to-line overvoltage with greater accuracy. VI. DISCUSSION A. Overvoltage Range The typical motor terminal overvoltages demonstrated experimentally and predicted by the developed model reached nearly two times of the dc-bus voltage. This basic phenomenon can be easily explained by transmission-line theory assuming propagation of a single pulse from cable to motor and mismatch of the terminating impedances. However, under some conditions, the motor terminal overvoltage can reach even higher levels. For example, as has been reported in [3], [4], [20], and [21],

10 2662 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 10, OCTOBER 2010 model to a single-phase DM model. Although the asymmetrical DM cable model gives the current in return path, once the model is available, it is possible to calculate the currents in the two parallel phases (conductors). However, it is not required for the purpose of studies considered herein. Fig. 21. Measured and predicted peak of line-to-line overvoltage for different cable lengths. the motor terminal overvoltage can reach three to four times the dc-bus voltage. Such very high level (higher than two times) of the motor terminal overvoltage may occur when, in addition to a long cable, the inverter operation produces double pulsing [3] or successive pulsing [4], pulsing polarity reversal [3] or bipolar pulsing [4], etc. Under such conditions, the preceding overvoltage transient has not fully decayed before the successive voltage pulses are injected by the inverter. The overlapped pulses traveling in the cable could result in higher terminal overvoltages. The contributing factors to this phenomenon include (but not limited to) drive modulation techniques, carrier frequency selected, cable natural frequency of oscillation, and cable high-frequency damping losses [3]. As has been pointed out in [3], special considerations and/or adjustments of the switching strategy should be used to avoid such conditions. The voltage stress may seriously degrade motor dielectric capability [32] and even causes motor winding insulation failures. B. Definition of Cable DM Depending on the numbers of cable internal conductors, the DM may be defined as a two-port or multiport network between/among two or more phase conductors [22]. As the focus of the paper is the line-to-line overvoltage transients, the DM assumed in this paper is defined by connecting two cable phases (conductors) in parallel and the remaining third phase becomes the returning path. It is also noted that the aforementioned twoport DM cable model is asymmetrical. Similar definition of the DM has been used in [4], [15], and [19] [21], and it is chosen here, since it corresponds to the physical path along which the IGBT-generated voltage pulse travels in the considered hardware system. The advantage of using the two-port DM network is that a single-phase TLM can be utilized [4], [15], [19] [21] to represent the DM cable characteristic, which further simplifies the three-phase cable modeling. Since the cable characteristics are directly measured in DM as a two-port network, the parameters of the single-phase TLM are directly derived from the measured DM characteristics. Therefore, there is no need to measure the three-phase coupling parameters and reducing the three-phase C. CM Overvoltage The CM overvoltage can also appear at motor terminals and create CM current through the stator-winding-to-frame and cable-to-ground parasitic capacitances. The CM current, as a high-frequency noise, could have harmful effects including the EMI for the surrounding equipments [33], and can also cause oil film dielectric breakdown inside the motor bearings and subsequently reduction of the bearing s lifetime [34] [37]. Interested reader can find an equivalent circuit for representing the bearingshaft path ([36], Fig. 2), which in general could be integrated with the CM equivalent circuit described in this paper. However, properly addressing this subject would require a dedicated publication and is beyond the scope of the paper. D. Factors Influencing the Accuracy of Cable Parameter Measurement It is noted that the accuracy of cable impedance measurement is potentially influenced by many factors including the length of the sample cable, measurement layout (i.e., cable layout with curves, straight lines, nearby conductors, grounding surfaces, proximity to magnetic materials, leads/fixture/adapter of the impedance analyzer, etc.). Therefore, a very careful measurement setup is designed such that the effect of these influencing factors is minimized. Also, the preferred sample cable length should not be too short [15], [20] [22] in order to be within the acceptable range of accuracy for a typical impedance analyzer. If a shorter cable length is used, the measured parameters could be too small, and therefore, less accurate. A 1-meter cable section was used for the measurements using the HP 4294 A analyzer with the type 16047E fixture/adapter [30], [38]. To minimize the influence of external parasitics, this fixture can directly clamp the wires of the cable segment under test without using additional leads. VII. CONCLUSION This paper has proposed a high-frequency modeling methodology for the long-cable-fed motor drive system to predict the overvoltage transients. An improved high-frequency motor model was proposed that accurately represents the highfrequency DM and CM impedance characteristics from hundreds of hertz to tens of megahertz. A frequency-dependent cable model is developed that includes the skin- and proximity effects as well as dielectric losses. The model parameters are easily identified from the measured DM impedance characteristics in a wide frequency range from 100 Hz to 10 MHz. The TLM technique was applied to the modeling of longcable-fed motor drive system in discrete time domain, which avoided the otherwise laborious formulation and integration of the overall system ordinary differential equations. The proposed

11 WANG et al.: HIGH-FREQUENCY MODELING OF THE LONG-CABLE-FED INDUCTION MOTOR DRIVE SYSTEM 2663 methodology is verified experimentally as well as compared against two conventional modeling techniques. It is shown that the proposed model represents an appreciable improvement in predicting the long-cable-fed motor drive overvoltage transients. ACKNOWLEDGMENT The authors would like to thank R. S. Y. Hui (City University of Hong Kong) for giving valuable comments in TLM modeling. APPENDIX A The motor high-frequency model parameters are calculated from the measured DM and CM characteristics as follows [17]: C g1 = 1 3 C HF C g2 = 1 3 (C total C HF ) (A1) (A2) L CM = ( 12π 2 C g2 f 2 Z 1) 1 (A3) R e 2 3 Z P R g1 = 2 3 Z Z 3 L d = L CM L DM R g2 1 3 Z Z 1 C t 1 6 (C g1 + C g2 ) (A4) (A5) (A6) (A7) (A8) L t = 1 ( ) 2 1 C t 2πfZ 2 2 (A9) R t = Z Z 2 cos (θ Z 2 ) (A10) L zu =3 ( 16π 2 C g1 f 2 Z 3) 1 (A11) APPENDIX B The cable high-frequency model parameters are calculated from the measured SC and OC characteristics as follows: L s1 = L SC-HF L s2 = L SC-LF L s1 R s1 = Z SC-LF cos (θ SC-LF) R s2 = Z SC-HF cos (θ SC-HF) R s1 C p1 = C OC-HF C p2 = C OC LF C p1 R p1 = Z OC-LF [cos (θ OC-LF)] 1 R p1//p2 = Z OC-HF [cos (θ OC-HF)] 1 R p2 =[(R p1//p2 ) 1 (R p1 ) 1 ] 1 (B1) (B2) (B3) (B4) (B5) (B6) (B7) (B8) (B9) REFERENCES [1] E. Persson, Transient effects in application of PWM inverters to induction motors, IEEE Trans. Ind. Appl., vol.28, no.5,pp ,Sep./Oct [2] L. A. 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Appl., vol. 33, no. 6, pp , Nov./Dec [38] Agilent LCR Meters Impedance Analyzers and Test Fixtures Selection Guide. (2009). Agilent Technologies Inc., Santa Clara, CA, USA. [Online]. Available: Liwei Wang (S 04) received the M.S. degree in electrical engineering from Tianjin University, Tianjin, China, and the Ph.D. degree in electrical and computer engineering from the University of British Columbia, Vancouver, BC in 2004 and 2010, respectively. He is currently a Postdoctoral Research Fellow in the Department of Electrical and Computer Engineering, University of British Columbia. From February 2009 to July 2009, he was an Internship Researcher at the ABB Corporate Research Center, Baden-Dättwil, Switzerland. His research interests include power system analysis and operation, electrical machine and drives, power electronic system and controls, renewable energy sources, and distributed generation. Carl Ngai-Man Ho (M 07) received the B.Eng. and M.Eng. degrees in 2002 and the Ph.D. degree in 2007 in electronic engineering from the City University of Hong Kong, Kowloon, Hong Kong. He was involved in research on the development of dynamic voltage regulation and restoration technology during his Ph.D. degree. From 2002 to 2003, he was a Research Assistant at the City University of Hong Kong. From 2003 to 2005, he was an Engineer at e.energy Technology Ltd., Hong Kong. In May 2007, he joined ABB Corporate Research Center, Baden-Dättwil, Switzerland, as a Scientist. His research interests include power electronics, power quality, modeling and control of power converters, and characterization of wide bandgap power semiconductor devices and applications of those. He holds two U.K. and one U.S. patents in the area of lighting applications. Dr. Ho was a Reviewer for the IEEE TRANSACTIONS ON POWER ELECTRONICS, the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, and several conferences. Francisco Canales (M 95) received the B.S. degree in mechanical and electrical engineering from the Universidad Veracruzana, Veracruz, Mexico, in 1989, the M.Sc. degree in electronic engineering from the Centro Nacional de Investigación y Desarrollo Tecnológico (CENIDET), Cuernavaca, Mexico, in 1994, and the Ph.D. degree in electrical engineering from the Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, in He was a Senior Research Assistant at the Center for Power Electronics Systems (CPES), Virginia Tech, where he was involved in core research and several industry-sponsored projects during his Ph.D. degree. He was an Associate Professor in the Department of Electronic Engineering, CENIDET. Currently he is a Principal Scientist at ABB Corporate Research Center, Baden-Dättwil, Switzerland, where he is working on high-density traction converters and industrial applications. Juri Jatskevich (M 99 SM 07) received the M.S.E.E. and the Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 1997 and 1999, respectively. Since 2002, he has been a Faculty Member at the University of British Columbia, Vancouver, where he is currently an Associate Professor of electrical and computer engineering. His research interests include power electronic systems, electrical machines and drives, average-value modeling and simulation. Dr. Jatskevich is currently a Chair of IEEE CAS Power Systems & Power Electronic Circuits Technical Committee, an Editor of the IEEE TRANSACTIONS ON ENERGY CONVERSION, the IEEE POWER ENGINEERING LETTERS, and an Associate Editor of the IEEE TRANSACTIONS ON POWER ELECTRONICS. He is also the Chair of the IEEE Task Force on Dynamic Average Modeling, under Working Group on Modeling and Analysis of System Transients Using Digital Programs.